Semiconductor device MESFET with upper and lower layers
RELATED APPLICATIONS 
     This is a continuation-in-part of our copending application Ser. No. 07/409,600 filed in Sept. 19, 1989, now abandoned for a Semiconductor Device. 
     DETAILED DESCRIPTION OF THE INVENTION 
     1. Background of the Invention 
     This invention relates to a semiconductor device using a GaAs metal-semiconductor field effect transistor (MESFET), specifically to that which can be used in machines and instruments requiring radiation hardness or radiation resistance. 
     2. Related Background Art 
     The devices which are used in aerospace systems and near nuclear furnaces are required to have high radiation hardness. The radiation includes gamma (γ) rays, neutron rays, proton rays, etc. Generally the gallium arsenide (GaAs) MESFETs and ICs based upon these FETs will withstand a total exposure dose of 1×10 8  roentgens with little if any change in characteristics. By contrast, silicon (Si) MOS circuits have failed at doses of 1×10 6  roentgens (Proceedings of Symposium of Space Development, 1987, ps. 35 to 38). 
     For improving the radiation hardness of the GaAs MESFET, the following devices have been proposed. In a first one, a p-type layer is buried below an n-type active layer to thereby decrease leakage current to the substrate, and the threshold voltage of a GaAs FET is improving in the radiation hardness. In a second one, the Schottky gate of a GaAs FET is shortened. 
     But these prior art devices have improved the radiation hardness up to a total exposure dose R of about 1×10 8  roentgens, but it cannot be said that these prior art devices have succeeded in attaining the sufficiently practical level (1×10 9  roentgens). Under these circumstances, no practical transistor having a radiation hardness level of about 1×10 9  roentgens has been realized. 
     On the other hand, Physics of Semiconductor Device, 2nd Edition, S.M. Sze., P. 334, describes a MESFET having two active layers. This MESFET has realized superior characteristics since the carrier concentration in the surface of the active layer lowers, and consequently the MESFET is free from the affects of a deletion layer. 
     Therefore an object of this invention is to provide a GaAs MESFET which has a simple structure and a high radiation hardness which has been improved by especially making use of changes of at least one of the threshold voltage, the saturated drain current, and the transconductance. 
     SUMMARY OF THE INVENTION 
     The inventors have noticed that when radiation is applied to a GaAs MESFET, the change amount ΔV th  of the threshold voltage V th  in the saturation region, the change rate α=I dssA  /I dss  of saturated drain current at normal gate bias I dss , and the change rate β=g mA  /g m  of a transconductance g m  has a constant relationship with the effective thickness t a  =t a1  +t a2  of the active layer and the change amount ΔN D  of the carrier concentrations N 1D , N 2D  and has found that the change amount ΔN D  has a constant quantitative relationship with the total exposure dose R. Based on this finding, they have completed this invention. 
     That is, this invention relates to a semiconductor device including a MESFET having an active layer doped in a GaAs crystal and a threshold voltage V th , the active layer comprising an upper layer having a carrier concentration N 1D  and an effective thickness t a1 , and a lower layer having a carrier concentration N 2D  and an effective thickness t a2  (N 1D&lt;N   2D ); and which normally operates when at least one of three conditions that a change amount ΔV th  of the threshold voltage V th  of the MESFET is within a tolerable change amount ΔV thL , a change rate α of the saturated drain current I dss  is within a tolerable change rate α L , and a change rate β of the transconductance g m  is a tolerable change rate β L  are satisfied, when ΔN D  represents a decrease in the amount of the carrier concentrations N 1D , N 2D  of the active layer due to the radiation exposure of a total dose R equal to or higher than 1×10 9  roentgens, μ and μ A  represent carrier mobilities in the active layer respectively before and after the radiation exposure, ε represents a dielectric constant of the active layer, and q represents an electron charge, at least one of the following three conditions 
     an effective thickness t a  =t a1  +t a2  of the active layer is given by 
     
         t.sub.a &lt;{(2ε·ΔV.sub.thL)/(q·ΔN.sub.D)}.sup.1/2 
    
     and, 
     a carrier concentration N 2D  of the lower layer of the active layer before the radiation exposure is given by 
     
         N.sub.2D &gt;ΔN.sub.D /{1-[α.sub.L (μ/μ.sub.A)].sup.1/2 }. 
    
     
         N.sub.2D &gt;ΔN.sub.D /{1-β.sub.L (μ/μ.sub.A)}. 
    
     In this arrangement, under irradiation of a total exposure dose even below 1×10 9  roentgens, at least any one of the threshold voltage V th , the saturated drain current I dss , and the transconductance g m  are within their corresponding preset ranges (tolerable ranges determined by the signal processing circuit). Consequently, the semiconductor device according to this invention comprising the GaAs MESFET and a signal processing circuit can operate normally as initially designed. 
     The present invention will become more fully understood from the detailed description given hereinbelow and the accompanying drawings which are given by way of illustration only, and thus are not to be considered as limiting the present invention. 
     Further scope of applicability of the present invention will become apparent from the detailed description given hereinafter. However, it should be understood that the detailed description and specific examples, while indication preferred embodiments of the invention, are given by way of illustration only, since various changes and modifications within the spirit and scope of the invention will become apparent to those skilled in the art from this detailed description. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a sectional view of a GaAs MESFET explaining the principle of this invention; 
     FIG. 2 is a graph showing the relationship of the total exposure dose R and a change amount ΔV th  of the threshold voltage of the MESFET involved in this invention; 
     FIG. 3 is a graph showing the relationship of the total exposure dose R and a decrease amount ΔN D  of the carrier concentration; 
     FIG. 4 is a graph showing the result of an experiment concerning the relation of the total exposure dose R and a change amount ΔV th  of the threshold voltage of a conventional MESFET; 
     FIG. 5 is a graph showing the relationship of the total exposure dose R dependence of a change rate α of the saturated drain current of the MESFET involved in this invention; 
     FIG. 6 is a graph showing the relationship of the total exposure dose R and a change of the saturated drain current I dss  ; 
     FIG. 7 is a graph showing the relationship of the total exposure dose R and a change rate β of the transconductance of the MESFET involved in this invention; 
     FIG. 8 is a graph showing the results of experiments on the relationship of the total exposure dose R and a change of the transconductance g m  of the conventional MESFET; 
     FIGS. 9 to 11 are characteristic curves of the threshold voltage, saturated drain current, and transconductance of the MESFET involved in this invention and of the conventional FETs for comparison in radiation hardness; and 
     FIG. 12 is a sectional view of a GaAs MESFET having an active layer in one layer. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     This invention will be explained in good detail with reference to the drawings showing the principle and structure of this invention. 
     The semiconductor device according to this invention comprises a GaAs MESFET, and a signal processing circuit cooperatively combined with the MESFET. The MESFET and the signal processing circuit can provide combination circuits, e.g., amplifiers, inverters, oscillators, digital logic arrays, etc. 
     FIG. 1 shows a sectional view of a GaAs MESFET having a recess gate structure. As shown in FIG. 1, an n-type active layer 2, and a heavily doped n-type (n +  -type) contact region 3 are formed on a semi-insulating GaAs substrate 1. The active layer 2 includes an upper layer 21 of a thickness t a1  and a lower layer 22 of a thickness t a2 . Parts of the n-type GaAs active layer 2 and the n +  -type contact region 3 for a gate region to be provided are etched off to form a recess structure. Then a source electrode 4 and a drain electrode 5 of ohmic metal are formed on the n +  -type contact region 3 by the vacuum evaporation. A gate electrode 6 of Schottky metal is formed on the n-type active layer 2. The part of the n-type active layer 2 directly below the gate electrode 6 has a sufficiently smaller thickness, compared with the n-type active layers of the conventional MESFETs, and the active layer 2 has a higher carrier concentration N 2D , compared with the carrier concentrations of the conventional MESFETs. 
     The MESFET in this combination circuit involved in this invention has a preset threshold voltage V th , a saturated drain current I dss  at normal gate bias, and a transconductance g m  in the saturation region. It has been known that their values change under radiation exposure. When a changed threshold voltage V thA , a changed saturated drain current I dssA , and a changed transconductance g mA  to which their initial values have changed due to radiation exposure are out of their preset ranges by the signal processing circuit, this combination circuit does not normally operate. Hereinafter in this specification, a tolerable value of a change amount ΔV th  of the threshold voltage V th  is represented by a tolerable change amount ΔV thL . A tolerable value of a change rate α=I dssA  /I dss  of the saturated drain current I dss  is represented by a tolerable change rate α L , and a tolerable value of a change rate α=g mA  /g m  of the transconductance is represented by a tolerable change rate β L . The values of ΔV thL , α L  and β L  vary depending on designs of the above described signal processing circuit, but generally ΔV thL  is equal to or lower than 0.2 V, and α L  and β L  are equal to or higher than 0.8. 
     As described above, it is known that the threshold voltage V th , etc. change under radiation exposure. Causes for these changes have been reported, firstly, decreases in a carrier concentration of the active layer due to radiation exposure, and secondly decreases in an electron mobility therein due to radiation exposure. The inventors discussed the first cause in detail and found the relationship 
     
         ΔN.sub.D =b·R.sup.c                         (1) 
    
     where b and c are constants holds between a decreased amount ΔN D  of a carrier concentrations N 1D , N 2D  and a total exposure dose R. Formula 1 holds when an initial carrier concentration N 2D  (before radiation exposure) of the active layer is 1×10 17  to 1×10 19  cm -3 , a total dose R of exposure radiation is 1×10 8  to 1×10 10  roentgens. The constants b and c have some ranges depending on changes of an initial carrier concentration of the active layer, qualities of the substrates, etc. 
     The values of the above-described constants b, c were given by the following two methods. In a first method, some samples of the GaAs MESFETs were prepared, and those samples were exposed to radiation to measure the change amounts ΔV th  of the threshold voltages. That is, since the relationship of Formula 5 which will be described below is given between the change amount ΔV th  and the carrier concentration decrease amount ΔN D , the relationship between the total dose R and the carrier concentration decrease amount ΔN D  can be given empirically by measuring the total dose R of radiation and the change amount ΔV th . Based on this relationship, the constants b, c in the above-described Formula 1 could be given. 
     According to the experiments conducted by the inventors, the constants b and c have ranges of 
     
         1.99×10.sup.10 ≦b≦3.98×10.sup.10 
    
     
         0.5≦c≦0.8 
    
     and the typical values are b=3.06×10 10 , c=0.678. Therefore, the typical value of the decreased amounts ΔN D  of the carrier concentration is defined by 
     
         ΔN.sub.D =3.06×10.sup.10 ·R.sup.0.678. 
    
     This first method is for measuring a change amount ΔV th  of the threshold voltage to calculate a carrier concentration decrease amount ΔN D  and accordingly for indirectly giving the relationship between the carrier concentration decrease amount ΔN D  and the total dose R. In contrast to the first method, in a second method, the relationship between the carrier concentration decrease amount ΔN D  and the total dose R is directly given. To this end, the Hall effect was measured. 
     First, a GaAs Hall element which has been made n-type by Si doping was prepared. The n-type GaAs layer was formed by epitaxial growth, and the carrier concentration distribution in the n-type GaAs layer is constant in the directions of the depth, length and width of the Hall element. A plurality of such samples having a 100 Å to 2 μm-thickness n-type GaAs layer and a 1×10 16  cm -3  to 5×10 18  cm -3  -carrier concentration were prepared. The Hall effect was measured on these samples to give the carrier concentration and Hall mobility. Then when a carrier concentration is represented by N, and a carrier concentration change amount is indicated by ΔN, a carrier extinction number is given by 
     
         ΔN=N(before radiation)-N(after radiation). 
    
     According to these experiments, the constants b, c have an allowance of 
     
         5×10.sup.5 ≦b≦1×10.sup.6 
    
     
         1.0≦c≦1.3. 
    
     The constant b is represented by b=6.65×10 5 , and the constant c is represented by c=1.17. Therefore, the representative value of the carrier concentration decrease amount ΔN D  is 
     
         ΔN.sub.D =6.65×10.sup.5 ·R.sup.1.17. 
    
     As seen from the above, the first indirect measuring method and the second direct measuring method give different values of the constants b, c. The reason for the occurrence of such differences is presumed as follows. That is, in the indirect measurement in accordance with the first method, on the assumption of the following (a), (b) and (c), the carrier concentration decrease amount ΔN D  was calculated based on the actually measured values of the change amount ΔV th . 
     (a) The depth profile of the effective carrier concentration is constant over the active layer of the GaAs MESFET. 
     (b) The decrease in the effective carrier concentration occurs uniformly over the active layer, and the thickness remains unchanged for whole irradiations. 
     (c) The carrier mobility is not changed by γ-ray radiation. 
     In contrast to this, in the direct measure in accordance with the second method, the carrier concentration decrease amount ΔN D  was measured actually by the Hall effect, and the above-described assumption was not used. It is considered that due to this, the difference in the values of the constants b, c between the first and the second methods took place. But anyway it is doubtless that the above-described Formula 1 is given. In the following description, the result of the second method, i.e., the measurement of the Hall effect, is taken as the values of the constants b, c. 
     The result of the measurement of the Hall effect is expressed in a two-dimensional logarithmic graph where γ-ray exposure dose is on the horizontal axis, and the carrier extinction number is on the vertical axis as shown by the dots in FIG. 3. The above-described Formula 1 expressed by 
     
         ΔN.sub.D =6.65×10.sup.5 ·R.sup.1.17 
    
     as described above is plotted as indicated by the dotted line in FIG. 3. 
     The carrier concentration decrease amount ΔN D  can be given by the above-described Hall effect measurement, but the general relationship between the change amount ΔV th  of the threshold voltage V th  of the GaAs MESFET and the carrier concentration decrease amount ΔN D  can be given theoretically as follows. 
     The theoretical value of the threshold voltage V th  of this GaAs MESFET is given by ##EQU1## given by solving the one dimensional Poisson equation 
     
         d.sup.2 φ.sub.1 /dχ.sup.2 =-qN.sub.1D /ε 
    
     
         (0≦χ≦t.sub.a1) 
    
     
         d.sup.2 φ.sub.2 /dχ.sup.2 =-qN.sub.2D /ε 
    
     
         (t.sub.a1 ≦χ≦t.sub.a1 +t.sub.a2) 
    
     under the boundary conditions of 
     
         dφ/dχ=0 at χ=t.sub.a1 +d(d&gt;t.sub.a2) 
    
     
         φ=-(V.sub.bi -V.sub.G) at χ=0. 
    
     In Formula 2, V bi  represents a built-in voltage of the MESFET; q, an electron charge; and ε, a dielectric constant of the n-type active layer 2. When the carrier concentrations N 1D , N 2D  of the n-type active layer 2 becomes N 1DA , N 2DA  due to radiation exposure, a threshold voltage V thA  after the radiation exposure is given by ##EQU2## Then, a change of the threshold voltage V th  due to the radiation exposure is given by ##EQU3## In Formula 4 ##EQU4## Therefore, when a decrease in the carrier concentration due to radiation exposure is given by 
     
         ΔN.sub.D =N.sub.D -N.sub.DA, 
    
     
         with t.sub.a =t.sub.a1 +t.sub.a2, 
    
     
         ΔV.sub.th ={(q·t.sub.a.sup.2)/2ε}·ΔN.sub.D(5). 
    
     In this Formula 5, the carrier concentration decreased amount ΔN D  can be given empirically by measuring Hall effect on the samples (Hall elements). Accordingly the theoretical value of the change amount ΔV th  of the threshold voltage can be given. Reversely, the change amount ΔV th  can be given experimentally using samples of the MESFET. 
     The inventors further studied the change amount ΔV th  of the threshold voltage V th  by irradiation by gamma rays in total exposure doses R=1×10 8  roentgens, 1×10 9  roentgens and 3×10 9  roentgens to a MESFET of FIG. 1 having the active layer 2 of a thickness of 450=(150+300) Å. The result is shown in FIG. 2 by black points. 
     Then, the above-described relationship given by measuring the Hall effect, 
     
         ΔN.sub.D =6.65×10.sup.5 ·R.sup.1.17 
    
     is adapted to the MESFET of FIG. 1 to give the change amount ΔV th  of the threshold voltage V th  using the above-described Formula 5. The result is as indicated by the dotted line in FIG. 2. The theoretical value of the change amount ΔV th  of the threshold voltage well agrees with the experimental value thereof. 
     In FIG. 2, with a total dose of R=1×10 9  roentgens, a change amount ΔV th  of the threshold voltage is as low as about 0.04 V. Therefore, it is confirmed that the radiation hardness is conspicuously improved when the active layer 2 has a thickness of about 450 Å. 
     Formula 1 described above was derived from actually measured values for six total doses of R=1×10 8  roentgens, 3×10 8  roentgens, 6×10 8  reoengens, 1×10 9  roentgens, 2×10 9  roentgens and 3×10 9  roentgens. It can be said that these values are insufficient data to derive a general formula. Then the inventors conducted a further experiment of irradiation by gamma rays from cobalt 60 to the GaAs MESFET of FIG. 12 which had the same geometrical structure except that in the active layer as the GaAs MESFET involved in this invention and has the active layer 2 in one layer. The active layer 2 had an effective thickness t a  of 1130 Å so that the carrier concentration N D  is 2.09×10 17  cm 3  . In this experiment, total exposure doses were R=1×10 6  roentgens, 1×10 7  roentgens, 1×10 8  roentgens, 3×10 8  roentgens, 1×10 9   roentgens, 2×10 9  roentgens and 3×10 9  roentgens. The resultant change amounts ΔV th  of the threshold voltage are shown in FIG. 4 by the black points. These points well agree with the theoretical values indicated by the dotted line. The change amounts ΔV th  after a radiation exposure of R=1×10 9  roentgens, however, was about 0.4 V, which was remarkably inferior to that in FIG. 2. 
     The results of these experiments on the threshold voltage show the following. Firstly, a major cause for the degradation of the threshold voltage of the MESFET due to radiation damage is a decrease in the carrier concentration of the active layer, and it was found that Formula 1 well expresses a decrease in the carrier concentration under the radiation exposure. Secondly, it has found that the change amount ΔV th  of the threshold voltage V th  can be set at a required value by setting only the thickness t a`=t   a1  +t a2  of the active layer. Specifically, as shown in FIG. 4, when the thickness t a  of the active layer 2 is set at about 1000 Å as in the conventional MESFETs, the radiation hardness is insufficient, but as shown in FIG. 2, when the thickness t a  =t a1  +t a2  is set at 450 Å, the radiation hardness is conspicuously improved. 
     Then, the inventors actually measured changes of the saturated drain current I dss  due to the radiation exposure of the same GaAs MESFETs as those which exhibited the threshold voltages V th  of FIGS. 2 and 4. As a result, the characteristic of the change rate α of the saturated drain current I dss  of FIG. 5 was obtained in use of same MESFET of FIG. 2. The change rate α of the saturated drain current I dss  of FIG. 6 was obtained in use of same MESFET of FIG. 4. In FIGS. 5 and 6, the black points indicate the experimental values, and the dotted lines indicate the theoretical values given by applying Formula 10 described below to Formula 1. 
     The theoretical formula for the change rate α=I dssA  /I dss  of the saturated drain current I dss  will be derived below. The saturated drain current I dss  of the MESFET is given for an intrinsic FET with a source resistance R s  kept out of consideration, by ##EQU5## In Formula 6, W g  represents a gate width; μ, an electron mobility in the active layer 2; L g , a gate length; and V G , a gate voltage. In Formula 6, N 1D  /N 2D  =0 based on N 1D  &lt;N 2D . Therefore, Formula 6 is rewritten d 1  =t a1 . When a saturated drain current I dss  for V G  =V bi  is represented by I DSS  for simplifying the computation, Formula 6 is rewritten into 
     
         I.sub.DSS ={(W.sub.G ·μ·q.sup.2 ·N.sub.2D)/(ε·L.sub.g)}·[t.sub.a1 ·t.sub.a2.sup.2 +(t.sub.a2 -t.sub.a1)t.sub.a2.sup.2 /2 -t.sub.a2.sup.3 /3]                                       (7) 
    
     when a saturated drain current after the radiation exposure is represented by I DSSA , a change rate α due to the radiation exposure is given based on Formula 7 by ##EQU6## where N 2DA  is a carrier concentration of the lower layer after the radiation exposure and is given by 
     
         N.sub.2DA =N.sub.2D -ΔN.sub.D                        (9). 
    
     Then Formula 8 is substituted by Formula 9 into 
     
         α{μ.sub.A (N.sub.2D -ΔN.sub.D).sup.2 }/(Δ·N.sub.2D.sup.2)                       (10). 
    
     Then Formula 10 will be discussed below. It is found that the change rate α is influenced by changes (μ⊖μ A ) of the electron mobility μ due to the radiation exposure. But μ A  /μ is around 0.95-0.98 when the carrier concentration before the radiation exposure is about 1×10 18  cm -3 . The change becomes smaller as the carrier concentration becomes higher. Then the computation was made with μ A  /μ=0.95. The results are shown by the dotted lines of FIGS. 5 and 6. As described above, it was confirmed that the results agreed with the experimental values. 
     These experiments on the saturated drain current I dss  and the studies of their results show the following. Firstly, a major cause for the degradation of the I dss  of the MESFET as total dose radiation affects is decreases of in the carrier concentration of the active layer, and it was found that Formula 1 is very descriptive of the decrease in the carrier concentration due to the radiation exposure. Secondly, the change rate α of the saturated drain current can be set at a required value by setting only the initial carrier concentration (before the radiation exposure) N 2D  of the lower layer 22 of the active layer, because μ is a constant in Formula 10, the value of μ A  /μ can be approximated, and ΔN D  can be determined depending on a total radiation dose in Formula 1. Specifically, when the carrier concentration N D  of the active layer 2 is set at about 2×10 17  cm -3  as in the conventional ones, the radiation hardness is insufficient as seen in FIG. 6. When the carrier concentration N 2D  of the lower layer is set at 1×10 18  cm -3 , the radiation hardness is outstandingly improved as seen in FIG. 5. 
     Then, the inventors actually measured changes of the transconductances g m  due to the radiation exposure in the saturation regions of the same GaAs MESFETs as those which exhibited the threshold voltages V th  of FIGS. 2 and 4 and the saturated drain current I dss  characteristics of FIGS. 5 and 6. As results, the change rate β of the transconductance g m  of FIG. 7 was obtained in use of same MESFET of FIG. 2 and 5, and the change rate β of the transconductance g m  of FIG. 8 was obtained in use of same MESFET of FIG. 4 and 6. In FIGS. 7 and 8, the black points indicate the experimental values, and the dotted lines indicate the theoretical values given by applying Formula 15 described below to Formula 1. 
     The theoretical formula for the change rate β=g mA  /g m  of the transconductance will be derived below. A transconductance g m  in the saturation region of the MESFET is given for an intrinsic FET with a source resistance R s  kept out of consideration, by ##EQU7## In this Formula, 
     
         d.sub.1 ={t.sub.a1.sup.2 -N.sub.1D ·t.sub.a1.sup.2 /N.sub.2D +2ε·(V.sub.bi -V.sub.G)/(q·N.sub.2D)}.sup.1/2(11). 
    
     In Formula 11, N 1D  /N 2D  =0 based on N 1D  &lt;N 2D . Therefore, Formula 11 is rewritten d 1  =t a1 . When a transconductance g m  for V G  =V bi  is represented by g mmax  for simplifying the computation, Formula 11 is rewritten into ##EQU8## 
     A change rate β due to the radiation exposure is given based on Formula 12 by ##EQU9## where N 2DA  is a carrier concentration of the lower layer after the radiation exposure and given by 
     
         N.sub.2DA =N.sub.2D -ΔN.sub.D                        (14). 
    
     Then Formula 13 is substituted by Formula 14 into 
     
         β={μ.sub.A (N.sub.2D -ΔN.sub.D)}/(μ·N.sub.2D)(15). 
    
     Then Formula 15 will be discussed below. It is found that the change rate β is influenced by changes (μ→μ A ) of the electron mobility μ due to the radiation exposure. But μ A  /μ is around 0.95 when the carrier concentration before the radiation exposure is about 1×10 18  cm -3  . The change becomes smaller as the carrier concentration becomes higher. Then the computation was made with μ A  /μ=0.95. The results were the dot lines of FIGS. 7 and 8. As described above, it was confirmed that the results agreed with the experimental values. 
     These experiments and the studies of their results show the following. Firstly, a major cause for the degradation of the transconductance of the MESFET as a total dose radiation affect is decreases of the carrier concentration in the active layer, and it was found that Formula 1 is very descriptive of the decrease of the carrier concentration due to the radiation exposure. Secondly, the change rate β of the transconductance can be set at a required value by setting only the initial carrier concentration N 2D  of the lower layer 22 of the active layer, because μ is a constant in Formula 15, the value of μ A  /μ can be approximated, and ΔN D  is determined based on Formula 1, depending on a radiation dose. Specifically, when the carrier concentration N D  of the active layer 2 is set at about 2×10 17  cm -3  as in the conventional ones, the radiation hardness is insufficient as seen in FIG. 8. When the carrier concentration N 2D  in the lower layer is set at 1×10 18  cm -3 , the radiation hardness is outstandingly improved as seen in FIG. 7. 
     It is possible that, based on the above described findings, a structure of a semiconductor device which is able to operate normally under the radiation exposure of a total dose R not only below 1×10 9  roentgens but also a total dose R above 1×10 9  roentgens can be specified based on a thickness t a  of the active layer 2 and a carrier concentration N 2D  in the lower layer 22. That is, in order that a GaAs MESFET and a signal processing circuit are combined into such semiconductor device, and the signal processing circuit can operate as designed when a tolerable change amount of the threshold voltage V th  of the MESFET is ΔV thL , an effective thickness t a  =t a1  +t a2  of the active layer 2 must be 
     
         t.sub.a &lt;{(2ε·ΔV.sub.thL)/(q·ΔN.sub.D)}.sup.1/2                                                        (16), 
    
     based on Formula 5. In this case, the carrier concentration N 2D  of the lower layer 22 of the active layer 2 is given as follows based on Formula 2 by 
     
         N.sub.2D ={[2ε/(q·t.sub.a.sup.2)]·(V.sub.bi -V.sub.th)}                                               (17) 
    
     Here, when a tolerable change amount ΔV thL  of the threshold voltage V th  for a total exposure dose of R=1×10 9  roentgens is specifically computed by 
     
         ΔV.sub.thL =0.1V(ΔV.sub.th &lt;0.1V), 
    
     a change amount ΔN D  of the carrier concentration is given by 
     
         ΔN.sub.D =2.25×10.sup.16 cm.sup.-3. 
    
     An effective thickness t a  =t a1  +t a2  of the active layer 2 is given below 529 Å based on Formula 16. Further, with a thickness of the active layer 2 set at below 529 Å, the carrier concentrations N 1D , N 2D  of the active layer is determined by their interrelationships with the thickness t a1 , t a2 . In this case, a dielectric constant of the active layer ##EQU10## an electron charge 
     
         q=1.602×10.sup.-19 C, 
    
     and a built-in voltage V bi  =0.7V. 
     In order that the combination circuit which relates to the semiconductor device operates as required when a tolerable change rate of the saturated drain current I dss  of the MESFET is represented by α L , an initial carrier concentration N 2D  of the lower layer 22 of the active layer 2 must be, based on Formula 10 by 
     
         N.sub.2D &gt;ΔN.sub.D /{1-[α.sub.L ( μ/μ.sub.A)].sup.1/2 }(18). 
    
     In this case, an effective thickness t a  =t a1  +t a2  of the active layer 2 is given by 
     
         t.sub.a ={[2ε/(q·N.sub.2D)](V.sub.bi -V.sub.th)}(19). 
    
     Here, for a total exposure dose of R=1×10 9  roentgens, with α L  (a tolerable change rate of the saturated drain current I DSS )=0.9 (I DSSA  &gt;0.9I DSS ), a change amount ΔN D  of the carrier concentration is given based on Formula 1 by 
     
         ΔN.sub.D =3.25×10.sup.16 cm.sup.-3, 
    
     and the carrier concentration N 2D  of the lower layer 22 of the active layer 2 becomes above 8.44×10 17  cm -3  based on Formula 10. 
     Further, in order that the GaAs MESFET and a signal processing circuit are combined into this semiconductor device, and the signal processing circuit can operate as designed when a tolerable change rate of the transconductance g m  in the saturation region of the MESFET is β L , an initial carrier concentration N 2D  of the lower layer 22 of the active layer 2 must be, based on Formula 15, 
     
         N.sub.2D &gt;ΔN.sub.D /{1-β.sub.L (μ/μ.sub.A)}(21). 
    
     In this case, an effective thickness t a  =t a1  +t a2  of the active layer 2 must be 
     
         t.sub.a ={[ε/(q·N.sub.2D ](V.sub.bi -V.sub.th)}.sup.1/2(22). 
    
     Here, for a total exposure dose of R=1×10 9  roentgens, with β L  (a tolerable change rate of the transconductance g mmax )=0.9(g mmaxA  &gt;0.9 g mmax ), a change amount ΔN D  is given by 
     
         ΔN.sub.D =2.25×10.sup.16 cm.sup.-3 
    
     based on Formula 1. The carrier concentration N 2D  of the lower layer of the active layer 2 is given above 4.50×10 17  cm -3  by Formula 15. 
     The semiconductor device according to this invention and the conventional ones will be compared in radiation hardness in FIG. 9 to 11. FIG. 9 shows change amounts ΔV th  of the threshold voltage V th  due to the radiation exposure. FIG. 10 shows change rates α of the saturated drain current I dss . FIG. 11 shows change rates β of the transconductance g m . In FIGS. 9 to 11, curves (a), (b) and (c) show characteristics of the conventional commercial MESFETs. The curve (b) corresponds to the characteristics of FIGS. 4, 6 and 8 for the active layer 2 of a 1130 Å effective thickness t a  and a carrier concentration of 2.09×10 17  cm -3 . The curve (d) shows the characteristics of a conventional HEMT (high electron mobility transistor). As evident from FIG. 9, these conventional semiconductor devices have change amounts ΔV th  of the threshold voltage as high as 0.2 to 0.3 V for a total dose of R=1×10 9  roentgens. The curve (e) in FIG. 9 shows the characteristic of a MESFET having a p-type layer buried below an n-type active layer for decreasing leakage current to the substrate, and the change amount ΔV th  is suppressed to about 0.12 V for R=1×10 9  rentgens. In contrast to this, in the MESFET according to this invention having the active layer 2 of an effective thickness t a  =t a1  +t a2  of 450 Å (corresponding to the characteristic of FIG. 2), the change amount ΔV th  is suppressed to a value lower than 0.1 V even for R=1×10 9  roentgens as indicated by the curve (f), and it is found that the radiation hardness is much improved. It is evident from FIGS. 10 and 11 that such improvement in the radiation hardness is also exhibited in the saturated drain current I dss  and the transconductance g m . 
     In this invention, even under the radiation exposure of a total exposure dose equal to or higher than R=1×10 8  roentgens, the values of the threshold voltage V th , saturated drain current I dss  and transconductance g m  remain within their tolerable ranges. A GaAs MESFET which is acknowledged as superior in radiation hardness characteristics must have radiation hardness for a total exposure dose of about 1.4×10 8  to 4.3×10 9  roentgens. For this exposure dose, the absorbed dose of GaAs is totally 1×10 8  to 3×10 9  rad. (1 roentgens=0.7 rads. in GaAs). On the other hand, the tolerable range of the change (positive shift) amount ΔV th  of the threshold voltage V th  is 0.2 V, and the tolerable ranges of the change rates α=I dss , β=g mA  /g m  of the saturated drain current I dss  and the transconductance g m  are about 80%. Specifically, when the change amount ΔV th  =0.15 V with a total exposure dose R=1.5×10 9   roentgens, it can be said that the GaAs MESFET has superior radiation hardness. 
     But what has to be noted here is that the above described tolerable change amount ΔV thL , and the tolerable change rates α L , β L  greatly vary depending on circuits combined with the GaAs MESFET. Specifically, one example is SCFL (Source coupled FET Logic) circuits which have low integrity but have enabled high speed operation. In theses circuits, the operation speed is substantially determined by a current flowing two transistors in the buffer stages. Accordingly, when values of the V th , I dss  and g m  vary due to a radiation exposure, the operation speed greatly varies. But the influence on the operation speed by the changes of values of V th , I dss  and g m  can be reduced by 1/3 to 1/4 by setting the values of resistors of the SCFL circuit at suitable values. Consequently, even in a SCFL circuit which allows for a change of the operation speed of only 10%, the tolerable change is 200 m/V for the threshold voltage V th  (V thL  =0.2 V), and the tolerable changes for the saturated drain current I dss  and the transconductance g m  are about 20% (α L  =0.8, β L  =0.8). 
     By contrast, another example is a memory cell for a memory IC which has high integrity on a semiconductor chip, the tolerable range for those changes are narrowed. Specifically, in this IC, a time in which one small memory cell charges and discharges the data lines occupies a large part of a total access time. Furthermore, each memory cell has transistors, resistors, etc. miniaturized for reducing power consumption. Consequently, the operation speed greatly varies depending on changes of the parameters. Specifically, in order to keep the change of the memory access time within 20%, the tolerable change amount of the threshold voltage V th  is only 50 mV ΔV thL  =0.05 V), and the tolerable change rates of the saturated drain current I dss  and the transconductance g m  are only 10% (α L  =0.9, β L  =0.9). 
     This invention is not limited to the above described embodiment and covers various modifications. 
     For example, the active layer is not necessarily formed by epitaxial growth but may be formed by ion implantation. The recess structure of FIG. 1 is not essential. 
     From the invention thus described, it will be obvious that the invention may be varied in many ways. Such variations are not to be regarded as a departure from the spirit and scope of the invention, and all such modifications as would be obvious to one skilled in the art are intended to be included within the scope of the following claims.